Abstract

This lesson is based on several interesting probability problems. Each problem has a somewhat unexpected answer; in fact, many people have a hard time accepting experimental results for these problems, as the results may seem counterintuitive. This very difference in expectations and actual results leads to a deeper consideration of the related mathematics and to acquiring new tools for solving problems, namely the ideas and formulas connected with conditional probability and probability of simultaneous events.

Objectives

Upon completion of this lesson, students will:

• have taken a closer look at conditional probability
• have learned the formula for probability of simultaneous independent events

Activities

• Racing Game with One Die
• Two Colors
• Monty Hall, 3 Doors

Standards

The activities and discussions in this lesson address the following Standards:

• Probability

Key Terms

• probability
• experimental probability
• theoretical probability
• conditional probability

Student Prerequisites

• Arithmetic: Students must be able to:
  • use addition, subtraction, multiplication and division to solve probability formulas
  • understand how tables can be used in multiplication
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

All activities in the lesson are better experienced by using the software, with individual students or small groups of students having enough time to explore the games and find answers to the related questions. If the activities have to be set up physically, the following materials are necessary (one set of materials for each group of students that will be doing the activity):

• access to a browser
• pencil and paper
• copies of the supplemental materials:
  • Racing Game Worksheet
  • Two Colors Worksheet
  • Monty Hall Worksheet
• For the Racing Game with One Die:
  1. one six-sided die
  2. The Racing Game Table to tally the results
• For the Two Colors game:
  1. three identical containers (e.g., small boxes or opaque cups)
  2. six objects of two different colors (three of each color), such as marbles or poker chips.
     The objects have to fit in the containers and have to be indistinguishable from each other by touch.
  3. The Two Colors Table to tally the results
• For the Monty Hall, 3 Doors activity:
  1. Three identical index cards
  2. The Monty Hall Table to tally the results

Lesson Outline

1. Begin by describing the Racing Game with One Die activity, which shows experimentally how the number of steps in the Racing Game affects the probability of winning.

   Groups of students can play the Racing Game with One Die either using the software (preferably) or rolling a six-sided die and using the Table to tally the results.

   Players in the game should have unequal chances to take a step. Knowing the probability of each player taking a step, students can try to predict the probability of each player winning the game, and try multiple experiments in order to test the prediction.

2. Lead a discussion about the Probability of Simultaneous Events to introduce the formula for probability of simultaneous independent events.

   This discussion is based on the results of the Racing Game with One Die. Each group of students can think about and discuss the following questions, later discussing them with other groups and with their mentor:

   1. The experimental probability of winning the game is not the same as the probability of taking one step. Why?
   2. What would happen to the probabilities if there were more than two steps to the finish?

3. Use the Two Colors game to perform experiments that will demonstrate conditional probability.

   There are three closed boxes. One box contains two green balls, another one contains two red balls and the last one has one red and one green ball. If students use the software, the computer will shuffle the boxes. If students use manipulatives, one of them should shuffle the boxes. A student chooses one box and picks one ball from it (without looking). If the first ball is red, the game starts over. If the first ball is green, the student wins if the second ball in the same box is also green.

   Groups of students can play the game many times, first trying to predict or guess their chances of winning, and keeping track of the results using the Table.

4. Next, initiate a discussion based on Conditional Probability.

   This discussion requires the active participation of the mentor. If there are students who want to take on the role of mentors, they can read the discussion ahead of time in order to prepare. This way discussions can happen in smaller groups.

Alternate Outlines

This lesson can be rearranged in several ways.

• Include the Monty Hall, 3 Doors activity to further clarify conditional probability. Each student or group of students can try to solve the problem and explain the solution. Then they can run the experiments on computers or by hand (in the latter case, recording the results in the Table), comparing experimental data with their solutions. Groups of students can discuss why their theoretical answers agree or do not agree with the data.
• Use the Think and check! discussion to help students understand the explanation of the Monty Hall problem and the Two Colors Game.
• Combine this lesson with the Unexpected Answers lesson.
• Or choose fewer of the activities to cover; for example, use only the Racing Game with One Die and the Conditional Probability discussion and make the focus conditional probability only. Use the Probability of Simultaneous Events discussion somewhere else in the Probability unit.
• Have students come up with their own version of the Two Colors game, and present their game and probability results to the class.

Suggested Follow-Up

After these discussions and activities, the students will have worked with condition probability and have seen the formula for the probability of simultaneous events. The next lesson, From Probability to Combinatorics and Number Theory, is devoted to data structures and their applications to probability theory. Tables and trees are introduced, and some of their properties are discussed.